Step by step
The question
Question: a sample of 36 has mean 40, and the SD of all the numbers is 6: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 40: the middle of our interval.
σ, the SD of all the numbers, is known: 6.
And n, the sample size, is 36.
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
1 Find z*
90 percent of the bell is in the middle.
That leaves 10 percent: 5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.9 plus 0.05 is 0.9500.
Now we look for 0.9500 inside the table.
0.9500 is halfway between 0.9495 and 0.9505.
So z is halfway between 1.64 and 1.65: 1.645.
2 Error bound
The error bound: z* times σ over the square root of n.
The square root of 36 is 6.
6/6 is 1.
Times 1.645: 1.645.
Careful!
We divide by the square root of 36, not by 36.
3 The interval
Take the error bound away from the middle: 40 minus 1.645 is 38.355.
And add it: 40 plus 1.645 is 41.645.
The 90 percent confidence interval is (38.355, 41.645).
4 Say it in words
In words: we are 90 percent confident that the true mean is between 38.355 and 41.645.
5 Check
Let's check: the middle of the interval should be 40.
38.355 plus 41.645, divided by 2, is 40.
Half the width should be the error bound: 41.645 minus 38.355, divided by 2, is 1.645.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.